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Monsieur Gauss [Gauss was still living when Hermite wrote this,
hence the polite 'Monsieur'] has opened up to us, Algebra and
the Theory of Numbers seem necessarily to be merged in the
same order of analytical concepts, of which our present know-
ledge does not yet permit us to form an accurate idea.*
He then makes a remark which, although not very clear, can
be interpreted as meaning that the key to the subtle connexions
between algebra, the higher arithmetic, and certain parts of the
theory of functions will be found in a thorough understanding
of what sort of 'numbers' are both necessary and sufficient for
the explicit solution of all types of algebraic equations. Thus,
for a?5  1 = 0, it is necessary and sufficient to understand
tf 1; for xs + ax + b = 0, where <z,5 are any given numbers,
what sort of a 'number' x must be invented in order that % may
be expressed explicitly in terms of a,b? Gauss of course gave one
kind of answer: any root x is a complex number* But this is only
a beginning. Abel proved that if only a finite number of rational
operations and extractions of roots are permitted, then there is
fzo explicit formula giving x in terms of a,6. We shall return to
this question later; Hermite even at this early date (184-8; he
was then twenty-six) seems' to have had one of his greatest
discoveries somewhere at the back of his head.
In his attitude toward numbers Hermite was somewhat of a
mystic in the tradition of Pythagoras and Descartes - the
latter's mathematical creed, as will appear in a moment, was
essentially Pythagorean. In other matters, too, the gentle
Hermite exhibited a marked leaning toward mysticism. Up to
the age of forty-three he was a tolerant agnostic, like so many
French men of science of his time. Then, in 1856, he fell suddenly
and dangerously ill. In this debilitated condition he was no
match for even the least persistent evangelist, and the ardent
Cauchy, who had always deplored his brilliant young friend's
open-mindedness on religious matters, pounced on the prostrate
Hermite and converted him to Roman Catholicism. Thence-
forth Hermite was a devout Catholic, and the practice of his
religion gave him much satisfaction.
Hermite's number-mysticism is harmless enough and it is one
of those personal things on which argument is futfle. Briefly,